Coxeter group structure of cosmological billiards on compact spatial manifolds

TL;DR

This paper systematically analyzes the Coxeter group structures of cosmological billiards in Einstein-p-form systems with compactified spatial manifolds, revealing how topology influences the billiard dynamics and their algebraic properties.

Contribution

It provides a comprehensive classification of Coxeter groups and billiard regions for all maximally oxidised theories under various compactifications, extending the understanding of cosmological billiards and their algebraic structures.

Findings

Coxeter groups may or may not be simplex groups depending on topology.

Billiard regions can be fundamental domains or galleries, affecting group presentations.

Chaotic behavior depends on the specific compactification and resulting Coxeter group.

Abstract

We present a systematic study of the cosmological billiard structures of Einstein-p-form systems in which all spatial directions are compactified on a manifold of nontrivial topology. This is achieved for all maximally oxidised theories associated with split real forms, for all possible compactifications as defined by the de Rham cohomology of the internal manifold. In each case, we study the Coxeter group that controls the dynamics for energy scales below the Planck scale as well as the relevant billiard region. We compare and contrast them with the Weyl group and fundamental domain that emerge from the general BKL analysis. For generic topologies we find a variety of possibilities: (i) The group may or may not be a simplex Coxeter group; (ii) The billiard region may or may not be a fundamental domain. When it is not a fundamental domain, it can be described as a sequence of pairwise adjacent chambers, known as a gallery, and the reflections in the billiard walls provide a non-standard presentation of the Coxeter group. We find that it is only when the Coxeter group is a simplex Coxeter group, and the billiard region is a fundamental domain, that there is a correspondence between billiard walls and simple roots of a Kac-Moody algebra, as in the general BKL analysis. For each compactification we also determine whether or not the resulting theory exhibits chaotic dynamics.